How to use the unit circle to find properties and identities of the sine and cosine functions? Grade 11 trigonometry questions are presented along with detailed Solutions and explanations.

A circle has an infinite number of symmetries with respects to lines through the center and a symmetry with respect to its center. We are interested here on the symmetries with respect to its center, the x-axis, the y-axis an the line y = x. It will be shown how the use of these symmetries allows us to write several identities in trigonometry.

Identities due to Symmetry of the Unit Circle on the origin, x and y axes

Four angles (θ, π - θ, π + θ and 2π - θ) are shown below in a unit circle. To each angle corresponds a point (A, B, C or D) on the unit circle.

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The four angles have the same reference angle equal to θ. Because of the symmetry of the circle, the four points form a rectangle ABCD as shown above. Points A and B are reflection of each other of the y-axis. Points A and C are reflection of each other on the origin of the system of axis. Points A and D are reflection of each other on the x-axis. Given the coordinates a and b of point A and using the symmetries of the circle, the coordinates of A, B, C and D are given by:

A: (a , b) , B: (- a , b), C: (- a , - b) and D: (a , - b)

We now express the coordinates of each point in terms of the sine and cosine of the corresponding angle as follows.

A: (a , b) = (cos θ , sin θ)

B: (- a , b) = (cos(π - θ) , sin(π - θ))

C: (- a , - b) = (cos(π + θ) , sin(π + θ))

D: (a , - b) = (cos(2π - θ) , sin(2π - θ))

Examples of Identities

Comparing the x and y-coordinates of points A and B, we can write

cos(π - θ) = - cos θ

sin(π - θ) = sin θ

Comparing the x and y-coordinates of points A and C, we can write

cos(π + θ) = - cos θ

sin(π + θ) = - sin θ

Comparing the x and y-coordinates of points A and D, we can write

cos(2π - θ) = cos θ

sin(2π - θ) = - sin θ

More Identities due to Symmetry of the Unit Circle on the x axis (Negative angles)

Two angles θ, and - θ are shown below in a unit circle to which correspond the points A and D on the unit circle.

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Points A and D are reflection of each other on the x-axis. Given the coordinates a and b of point A, the coordinates of D are given by:

D: (a , - b)

We now express the coordinates of points A and D in terms of the sine and cosine of the corresponding angle as follows.

A: (a , b) = (cos θ , sin θ)

D: (a , - b) = (cos(- θ) , sin(- θ))

Examples of Identities that may be Deduced

cos(- θ) = cos θ

sin( - θ) = - sin θ

Identities due to Symmetry of the Unit Circle on the line y = x

Points A and B shown in the unit circle below are reflection of each other on the line y = x. Because of the symmetry of the unit circle with respect to the line y = x, the corresponding angles to these points are θ and π/2 - θ as shown below.

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Points A and B are reflection of each other on the line y = x. Given the coordinates a and b of point A, the coordinates of B are given by:

B: (b , a)

We now express the coordinates of points A and B in terms of the sine and cosine of the corresponding angle as follows.